<p>The authors parallelize the embedding strategy for mapping any two-dimensional grid into its optimal hypercube with minimal dilation. The parallelization allows each hypercube node to independently determine, in constant time, which grid node it will simulate and the communication paths it will take to reach the hypercube nodes that simulate its grid-neighbors. The paths between grid-neighbors are chosen in such a way as to curb the congestion at each hypercube node and across each hypercube edge. Explicity, the node congestion for the embedding is at most 6, one above optimal, while the edge congestion is at most 5.</p>